Whittle–Matérn fields with general smoothness

Introduction

In this vignette we will introduce how to fit Whittle–Matérn fields with general smoothness based on finite element and rational approximations. The theory for this approach is provided in Bolin et al. (2023) and Bolin, Simas, and Xiong (2023). For the implementation, we make use of the rSPDE package for the rational approximations.

These models are thus implemented using finite element approximations. Such approximations are not needed in for integer smoothness parameters, and for the details about the exact models we refer to the vignettes

• Whittle–Matérn fields on metric graphs
• INLA and inlabru interfaces

For details on the construction of metric graphs, see Working with metric graphs

For further details on data manipulation on metric graphs, see Data manipulation on metric graphs

Constructing the graph and the mesh

We begin by loading the rSPDE and MetricGraph packages:

  library(rSPDE)
  library(MetricGraph)

As an example, we consider the following metric graph

  edge1 <- rbind(c(0,0),c(1,0))
  edge2 <- rbind(c(0,0),c(0,1))
  edge3 <- rbind(c(0,1),c(-1,1))
  theta <- seq(from=pi,to=3*pi/2,length.out = 20)
  edge4 <- cbind(sin(theta),1+ cos(theta))
  edges = list(edge1, edge2, edge3, edge4)
  graph <- metric_graph$new(edges = edges)
  graph$plot()

To construct a FEM approximation of a Whittle–Matérn field with general smoothness, we must first construct a mesh on the graph.

  graph$build_mesh(h = 0.5)
  graph$plot(mesh=TRUE)

In the command build_mesh, the argument h decides the largest spacing between nodes in the mesh. As can be seen in the plot, the mesh is very coarse, so let’s reduce the value of h and rebuild the mesh:

graph$build_mesh(h = 0.01)

We are now ready to specify the model \[ (\kappa^2 - \Delta)^{\alpha/2} \tau u = \mathcal{W} \] for the Whittle–Matérn field \(u\). For this, we use the matern.operators function from the rSPDE package:

  sigma <- 1.3
  range <- 0.15
  nu <- 0.8 

  rspde.order <- 2
  op <- matern.operators(nu = nu, range = range, sigma = sigma, 
                         parameterization = "matern",
                         m = rspde.order, graph = graph)                     

As can be seen in the code, we specify \(\kappa\) via the practical correlation range \(\sqrt{8\nu}/\kappa\). Also, the model is not parametrized by \(\tau, \alpha\) but instead by \(\sigma, \nu\). Here, sigma denotes the standard deviation of the field and nu is the smoothness parameter, which is related to \(\alpha\) via the relation \(\alpha = \nu + 1/2\). The object op_cov contains the matrices needed for evaluating the distribution of the stochastic weights in the FEM approximation.

Let us simulate the field \(u\) at the mesh locations and plot the result:

u <- simulate(op)
graph$plot_function(X = u, plotly = TRUE)

If we want to evaluate \(u(s)\) at some locations \(s_1,\ldots, s_n\), we need to multiply the weights with the FEM basis functions \(\varphi_i(s)\) evaluated at the locations. For this, we can construct the observation matrix \(\boldsymbol{\mathrm{A}}\), with elements \(A_{ij} = \varphi_j(s_i)\), which links the FEM basis functions to the locations. This can be done by the function fem_basis in the metric graph object. To illustrate this, let us simulate some observation locations on the graph and construct the matrix:

obs.per.edge <- 100
obs.loc <- NULL
for(i in 1:graph$nE) {
  obs.loc <- rbind(obs.loc,
                   cbind(rep(i,obs.per.edge), runif(obs.per.edge)))
}
n.obs <- obs.per.edge*graph$nE
A <- graph$fem_basis(obs.loc)

In the code, we generate \(50\) observation locations per edge in the graph, drawn at random. It can be noted that we assume that the observation locations are given in the format \((e, d)\) where \(e\) denotes the edge of the observation and \(d\) is the position on the edge, i.e., the relative distance from the first vertex of the edge.

To compute the precision matrix from the covariance-based rational approximation one can use the precision() method on object returned by the matern.operators() function:

  Q <- precision(op)

As an illustration of the model, let us compute the covariance function between the process at \(s=(2,0.1)\), that is, the point at edge 2 and distance on edge 0.1, and all the other mesh points. To this end, we can use the helper function cov_function_mesh that is contained in the op object:

  c_cov <- op$cov_function_mesh(matrix(c(2,0.1),1,2))
  graph$plot_function(c_cov, plotly = TRUE)

Using the model for inference

There is built-in support for computing log-likelihood functions and performing kriging prediction in the rSPDE package which we can use for the graph model. To illustrate this, we use the simulation to create some noisy observations of the process. We generate the observations as \(Y_i = 1 + 2x_{i1} - 3 x_{i2} u(s_i) + \varepsilon_i\), where \(\varepsilon_i \sim N(0,\sigma_e^2)\) is Gaussian measurement noise, \(x_1\) and \(x_2\) are covariates generated the relative positions of the observations on the graph.

    sigma.e <- 0.1

    x1 <- obs.loc[,1]
    x2 <- obs.loc[,2]

    Y <- 1 + 2*x1 - 3*x2 + as.vector(A %*% u + sigma.e * rnorm(n.obs))

Let us now fit the model. To this end we will use the graph_lme() function (that, for the finite element models, acts as a wrapper for the rspde_lme() function from the rSPDE package). To this end, let us now assemble the data.frame() with the observations, the observation locations and the covariates:

df_data <- data.frame(y = Y, edge_number = obs.loc[,1],
                        distance_on_edge = obs.loc[,2],
                        x1 = x1, x2 = x2)

Let us now add the data to the graph object and plot it:

graph$add_observations(data = df_data, normalized = TRUE)

graph$plot(data = "y")

We can now fit the model. To this end, we use the graph_lme() function and set the model to 'WM’.

fit <- graph_lme(y ~ x1 + x2, graph = graph, model = "WM")

Let us obtain a summary of the model:

summary(fit)

## 
## Latent model - Whittle-Matern
## 
## Call:
## graph_lme(formula = y ~ x1 + x2, graph = graph, model = "WM")
## 
## Fixed effects:
##             Estimate Std.error z-value Pr(>|z|)    
## (Intercept)   0.4326    0.7208   0.600  0.54843    
## x1            2.1130    0.2021  10.456  < 2e-16 ***
## x2           -2.3626    0.7190  -3.286  0.00102 ** 
## 
## Random effects:
##        Estimate Std.error z-value
## alpha  1.254255  0.023625  53.091
## tau    0.057719  0.006288   9.179
## kappa 15.560492  2.454061   6.341
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu     0.75426   0.02362  31.927
## sigma  1.34735   0.13608   9.901
## range  0.15786   0.02365   6.676
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev 0.097240  0.006306   15.42
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -126.9842 
## Number of function calls by 'optim' = 183
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  20.71652 secs

We can also obtain additional information by using the function glance():

glance(fit)

## # A tibble: 1 × 9
##    nobs  sigma logLik   AIC   BIC deviance df.residual model                                      alpha
##   <int>  <dbl>  <dbl> <dbl> <dbl>    <dbl>       <dbl> <chr>                                      <dbl>
## 1   400 0.0972  -127.  268.  296.     254.         393 Covariance-Based Matern SPDE Approximation  1.25

Let us compare the values of the parameters of the latent model with the true ones:

print(data.frame(sigma = c(sigma, fit$matern_coeff$random_effects[2]), 
                   range = c(range, fit$matern_coeff$random_effects[3]),
                   nu = c(nu, fit$matern_coeff$random_effects[1]),
                   row.names = c("Truth", "Estimates")))

##              sigma     range        nu
## Truth     1.300000 0.1500000 0.8000000
## Estimates 1.347354 0.1578632 0.7542551

  df_pred <- data.frame(edge_number = graph$mesh$VtE[,1],
                        distance_on_edge = graph$mesh$VtE[,2],
                        x1 = graph$mesh$VtE[,1],
                        x2 = graph$mesh$VtE[,2])

  u.krig <- predict(fit, newdata = df_pred, normalized = TRUE)

  graph$plot_function(as.vector(u.krig$mean))  

p <- augment(fit, newdata = df_pred, normalized = TRUE) %>% 
          graph$plot_function(data = ".fitted", plotly = TRUE)

graph$plot(data = "y", p = p, plotly = TRUE)          

Fitting a model with replicates

Let us now illustrate how to simulate a data set with replicates and then fit a model to such data. To simulate a latent model with replicates, all we do is set the nsim argument to the number of replicates.

  n.rep <- 30
  u.rep <- simulate(op, nsim = n.rep)

Now, let us generate the observed values \(Y\):

  sigma.e <- 0.3
  Y.rep <- A %*% u.rep + sigma.e * matrix(rnorm(n.obs * n.rep), ncol = n.rep)

Note that \(Y\) is a matrix with 20 columns, each column containing one replicate. We need to turn y into a vector and create an auxiliary vector repl indexing the replicates of y:

y_vec <- as.vector(Y.rep)
repl <- rep(1:n.rep, each = n.obs)                       

df_data_repl <- data.frame(y = y_vec,
                              edge_number = rep(obs.loc[,1], n.rep),
                              distance_on_edge = rep(obs.loc[,2], n.rep), 
                              repl = repl)

Let us clear the previous observations and add the new data to the graph:

graph$add_observations(data = df_data_repl, normalized = TRUE, 
                            group = "repl", clear_obs = TRUE)

We can now fit the model in the same way as before by using the rspde_lme() function. Note that we can optimize in parallel by setting parallel to TRUE. If we do not specify which replicate to consider, in the which_repl argument, all replicates will be considered.

fit_repl <- graph_lme(y ~ -1, graph = graph, model = "WM", parallel = TRUE)

## Warning in rSPDE::rspde_lme(formula = formula, loc = cbind(df_data[[".edge_number"]], : The optimization failed to provide a numerically positive-definite Hessian. You can try to obtain a positive-definite Hessian by setting 'improve_hessian' to TRUE or by setting 'parallel' to FALSE, which allows other optimization methods to be
## used.

## Warning in sqrt(diag(inv_fisher)): NaNs produced

Observe that we have received a warning saying that the Hessian was not positive-definite, which ended up creating NaNs for the standard errors. Indeed, let us see a summary of the fit:

summary(fit_repl)

## 
## Latent model - Whittle-Matern
## 
## Call:
## graph_lme(formula = y ~ -1, graph = graph, model = "WM", parallel = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##       Estimate Std.error z-value
## alpha  1.27025       NaN     NaN
## tau    0.05686       NaN     NaN
## kappa 15.19324   0.43202   35.17
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.770251       NaN     NaN
## sigma 1.324418  0.025423   52.09
## range 0.163384  0.005007   32.63
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev 0.300734  0.002942   102.2
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -9741.773 
## Number of function calls by 'optim' = 66
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  13.66658 secs 
##   set up the parallelization = 5.48799 secs

Let us, then, follow the suggestion from the warning and refit the model setting improve_hessian to TRUE. This will obtain a more precise estimate of the Hessian, which can possibly fix this issue:

fit_repl <- graph_lme(y ~ -1, graph = graph, model = "WM", 
                      parallel = TRUE, improve_hessian = TRUE)

We see that we did not receive any warning now, and the Std. errors were computed accordingly:

summary(fit_repl)

## 
## Latent model - Whittle-Matern
## 
## Call:
## graph_lme(formula = y ~ -1, graph = graph, model = "WM", parallel = TRUE, 
##     improve_hessian = TRUE)

## 
## No fixed effects.

## 
## Random effects:
##       Estimate Std.error z-value
## alpha  1.27025   0.09193  13.817
## tau    0.05686   0.02264   2.512
## kappa 15.19324   2.34526   6.478
## 
## Random effects (Matern parameterization):
##       Estimate Std.error z-value
## nu    0.770251  0.091932   8.379
## sigma 1.324418  0.025423  52.094
## range 0.163384  0.005007  32.629
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev 0.300734  0.004159   72.31
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -9741.773 
## Number of function calls by 'optim' = 66
## Optimization method used in 'optim' = L-BFGS-B
## 
## Time used to:     fit the model =  12.60502 secs 
##   compute the Hessian = 6.58939 secs 
##   set up the parallelization = 5.58605 secs

Let us also take a glance of the fit:

glance(fit_repl)

## # A tibble: 1 × 9
##    nobs sigma logLik    AIC    BIC deviance df.residual model                                      alpha
##   <int> <dbl>  <dbl>  <dbl>  <dbl>    <dbl>       <dbl> <chr>                                      <dbl>
## 1 12000 0.301 -9742. 19492. 19521.   19484.       11996 Covariance-Based Matern SPDE Approximation  1.27

Let us compare the values of the parameters of the latent model with the true ones:

print(data.frame(sigma = c(sigma, fit_repl$matern_coeff$random_effects[2]), 
                   range = c(range, fit_repl$matern_coeff$random_effects[3]),
                   nu = c(nu, fit_repl$matern_coeff$random_effects[1]),
                   row.names = c("Truth", "Estimates")))

##              sigma     range       nu
## Truth     1.300000 0.1500000 0.800000
## Estimates 1.324418 0.1633845 0.770251

Let us do kriging. We will use the same prediction locations as in the previous example. Let us get prediction for replicate 10, then add the original observations on top of them:

p <- augment(fit_repl, which_repl = 10, newdata = df_pred, normalized = TRUE) %>% 
          graph$plot_function(data = ".fitted", plotly = TRUE)

graph$plot(data = "y", group = 10, plotly = TRUE, p = p)

Using the R-INLA implementation

We also have an R-INLA implementation of the rational SPDE approach for metric graphs.

We begin by defining the model by using the rspde.metric_graph() function. This function contains the same arguments as the function rspde.matern(). We refer the reader to the R-INLA implementation of the rational SPDE approach vignette for further details.

We begin by clearing the previous observations and adding the observations (for the case without replicates) to the graph:

graph$clear_observations()
graph$add_observations(data = df_data, normalized = TRUE)

Let us create the model object:

  library(INLA)
  rspde_model <- rspde.metric_graph(graph)

By default, the order of the rational approximation is 2.

We can now create the auxiliary quantities that will be needed with the graph_data_rspde() function:

  data_rspde <- graph_data_rspde(rspde_model, name = "field")

The remaining is standard: we create the formula object, the stack object, and then fit the model by using the inla() function. So, first we create the formula object:

  f.s <- y ~ -1 + Intercept + x1 + x2 + f(field, model = rspde_model)

Now we create the inla.stack object. To such an end, observe that data_rspde contains the dataset as the data component, the index as the index component and the so-called A matrix as the basis component. We will now create the stack using these components:

  stk.dat <- inla.stack(
    data = data_rspde[["data"]]["y"], A = list(data_rspde[["basis"]],1), tag = "est",
    effects =
      list(c(
        data_rspde[["index"]],
        list(Intercept = 1)), list(x1 = data_rspde[["data"]]["x1"] ,
                                      x2 = data_rspde[["data"]]["x2"])
      )
    )

Finally, we can fit the model:

  rspde_fit <- inla(f.s, data = inla.stack.data(stk.dat),
    control.inla = list(int.strategy = "eb"),
    control.predictor = list(A = inla.stack.A(stk.dat), compute = TRUE)
  )

We can use the same functions as the rspde fitted models in inla. For instance, we can see the results in the original scale by creating the result object:

  result_fit <- rspde.result(rspde_fit, "field", rspde_model)
  summary(result_fit)

##             mean        sd 0.025quant 0.5quant 0.975quant     mode
## std.dev 1.436510 0.1576160   1.148280 1.429490   1.766580 1.417510
## range   0.171434 0.0373817   0.106204 0.169007   0.252029 0.164629
## nu      0.763701 0.1033980   0.579195 0.756731   0.983002 0.736587

Let us compare with the true values:

  result_df <- data.frame(
    parameter = c("std.dev", "range", "nu"),
    true = c(sigma, range, nu),
    mean = c(
      result_fit$summary.std.dev$mean,
      result_fit$summary.range$mean,
      result_fit$summary.nu$mean
    ),
    mode = c(
      result_fit$summary.std.dev$mode,
      result_fit$summary.range$mode,
      result_fit$summary.nu$mode
    )
  )
  print(result_df)

##   parameter true      mean      mode
## 1   std.dev 1.30 1.4365139 1.4175077
## 2     range 0.15 0.1714338 0.1646293
## 3        nu 0.80 0.7637009 0.7365873

We can also plot the posterior marginal densities with the help of the gg_df() function:

  posterior_df_fit <- gg_df(result_fit)

  library(ggplot2)

  ggplot(posterior_df_fit) + geom_line(aes(x = x, y = y)) + 
  facet_wrap(~parameter, scales = "free") + labs(y = "Density")

  pred_loc <- graph$mesh$VtE

graph$add_observations(data = data.frame(y=rep(NA,nrow(pred_loc)), 
                                x1 = graph$mesh$VtE[,1],
                                x2 = graph$mesh$VtE[,2],
                                edge_number = pred_loc[,1], 
                                distance_on_edge = pred_loc[,2]), 
                                normalized = TRUE)

  rspde_model_prd <- rspde.metric_graph(graph) 
  data_rspde_prd <- graph_data_rspde(rspde_model_prd, only_pred = TRUE)

  ef.prd <- 
    list(c(data_rspde_prd[["index"]], list(Intercept = 1)), 
          list(x1 = data_rspde_prd[["data"]][["x1"]],
                x2 = data_rspde_prd[["data"]][["x2"]]))
  stk.prd <- inla.stack(
    data = data.frame(y = data_rspde_prd[["data"]][["y"]]),
    A = list(data_rspde_prd[["basis"]],1), tag = "prd",
    effects = ef.prd
  )
  stk.all <- inla.stack(stk.dat, stk.prd)

rspde_fitprd <- inla(f.s,
  data = inla.stack.data(stk.all),
  control.predictor = list(
    A = inla.stack.A(stk.all),
    compute = TRUE, link = 1
  ),
  control.compute = list(
    return.marginals = FALSE,
    return.marginals.predictor = FALSE
  ),
  control.inla = list(int.strategy = "eb")
)

id.prd <- inla.stack.index(stk.all, "prd")$data
m.prd <- rspde_fitprd$summary.fitted.values$mean[id.prd]

  graph$plot_function(m.prd, plotly = TRUE)  

Using R-INLA implementation to fit models with replicates

Let us begin by cloning the graph and clearing the observations on the cloned graph:

graph_rep <- graph$clone()
graph_rep$clear_observations()

We will now add the data with replicates to the graph:

graph_rep$add_observations(data = data.frame(y=as.vector(Y.rep), 
                          edge_number = rep(obs.loc[,1], n.rep), 
                          distance_on_edge = rep(obs.loc[,2], n.rep),
                          repl = rep(1:n.rep, each = n.obs)), 
                          group = "repl",
                          normalized = TRUE)

Let us create a new rspde model object:

rspde_model_rep <- rspde.metric_graph(graph_rep)

To fit the model with replicates we need to create the auxiliary quantities with the graph_data_rspde() function, where we set the repl argument in the function graph_data_spde to .all since we want to use all replicates:

data_rspde_rep <- graph_data_rspde(rspde_model_rep, name = "field", repl = ".all")

Let us now create the corresponding inla.stack object:

st.dat.rep <- inla.stack(
  data = data_rspde_rep[["data"]],
  A = data_rspde_rep[["basis"]],
  effects = data_rspde_rep[["index"]]
)

Observe that we need the response variable y to be a vector. We can now create the formula object, remembering that since we gave the name argument field, when creating the index, we need to pass field.repl to the formula:

f.rep <-
  y ~ -1 + f(field,
    model = rspde_model_rep,
    replicate = field.repl
  )

We can, finally, fit the model:

rspde_fit_rep <-
  inla(f.rep,
    data = inla.stack.data(st.dat.rep),
    family = "gaussian",
    control.predictor =
      list(A = inla.stack.A(st.dat.rep))
  )

We can obtain the estimates in the original scale with the rspde.result() function:

  result_fit_rep <- rspde.result(rspde_fit_rep, "field", rspde_model_rep)
  summary(result_fit_rep)

##             mean         sd 0.025quant 0.5quant 0.975quant     mode
## std.dev 1.337860 0.03153290   1.269220 1.341100   1.390940 1.353590
## range   0.174413 0.00716223   0.159044 0.175042   0.186715 0.177525
## nu      0.729762 0.02390070   0.689765 0.727190   0.782391 0.718913

Let us compare with the true values of the parameters:

  result_rep_df <- data.frame(
    parameter = c("std.dev", "range", "nu"),
    true = c(sigma, range, nu),
    mean = c(
      result_fit_rep$summary.std.dev$mean,
      result_fit_rep$summary.range$mean,
      result_fit_rep$summary.nu$mean
    ),
    mode = c(
      result_fit_rep$summary.std.dev$mode,
      result_fit_rep$summary.range$mode,
      result_fit_rep$summary.nu$mode
    )
  )
  print(result_rep_df)

##   parameter true      mean      mode
## 1   std.dev 1.30 1.3378611 1.3535921
## 2     range 0.15 0.1744129 0.1775250
## 3        nu 0.80 0.7297618 0.7189134

We can also plot the posterior marginal densities with the help of the gg_df() function:

  posterior_df_fit_rep <- gg_df(result_fit_rep)

  ggplot(posterior_df_fit_rep) + geom_line(aes(x = x, y = y)) + 
  facet_wrap(~parameter, scales = "free") + labs(y = "Density")

Using inlabru implementation

The inlabru package allows us to fit models and do kriging in a straighforward manner, without having to handle A matrices, indices nor inla.stack objects. Therefore, we suggest the reader to use this implementation when using our implementation to fit real data.

Let us clear the graph, since it contains NA observations we used for prediction, add the observations again, and create a new rSPDE model object:

graph$clear_observations()
graph$add_observations(data = df_data, 
                          normalized = TRUE)
rspde_model <- rspde.metric_graph(graph)

Let us now load the inlabru package and create the component (which is inlabru’s formula-like object). Since we are using the data from the graph, inlabru will also obtain the locations from the graph, thus, there is no need to provide the locations. However, we need a name for the locations for using inlabru’s predict method. Therefore, we can choose any name for the location that is not a name being used in the graph’s data. In our case we will use the name loc:

    library(inlabru)

## Loading required package: fmesher

    cmp <-
    y ~ -1 + Intercept(1) + x1 + x2 + field(loc, 
                          model = rspde_model)                   

Let us now build the auxiliary data to be used with the graph_data_rspde() function, where we pass the name of the location variable in the above formula as the loc_name argument, which in this case is "loc":

data_rspde_bru <- graph_data_rspde(rspde_model, loc_name = "loc")

Now, we can directly fit the model of the graph_data_spde() function:

  rspde_bru_fit <-
    bru(cmp,
        data=data_rspde_bru[["data"]],
      options=list(
      family = "gaussian")
    )

Let us now obtain the estimates of the parameters in the original scale by using the rspde.result() function:

  result_bru_fit <- rspde.result(rspde_bru_fit, "field", rspde_model)
  summary(result_bru_fit)

##             mean        sd 0.025quant 0.5quant 0.975quant     mode
## std.dev 1.497600 0.1275290   1.301190 1.477870   1.794350 1.432310
## range   0.218502 0.0587804   0.140527 0.205503   0.366833 0.182656
## nu      0.670926 0.0839861   0.492888 0.678237   0.816284 0.701602

Let us compare with the true values of the parameters:

  result_bru_df <- data.frame(
    parameter = c("std.dev", "range", "nu"),
    true = c(sigma, range, nu),
    mean = c(
      result_bru_fit$summary.std.dev$mean,
      result_bru_fit$summary.range$mean,
      result_bru_fit$summary.nu$mean
    ),
    mode = c(
      result_bru_fit$summary.std.dev$mode,
      result_bru_fit$summary.range$mode,
      result_bru_fit$summary.nu$mode
    )
  )
  print(result_bru_df)

##   parameter true      mean      mode
## 1   std.dev 1.30 1.4975951 1.4323144
## 2     range 0.15 0.2185020 0.1826565
## 3        nu 0.80 0.6709262 0.7016016

We can also plot the posterior marginal densities with the help of the gg_df() function:

  posterior_df_bru_fit <- gg_df(result_bru_fit)

  ggplot(posterior_df_bru_fit) + geom_line(aes(x = x, y = y)) + 
  facet_wrap(~parameter, scales = "free") + labs(y = "Density")

  data_prd_list <- graph$get_mesh_locations(bru = TRUE,
                                            loc = "loc")
  data_prd_list[["x1"]] <- data_prd_list$loc[,1]
  data_prd_list[["x2"]] <- data_prd_list$loc[,2]

  y_pred <- predict(rspde_model, cmp, rspde_bru_fit, newdata=data_prd_list, ~Intercept + x1 + x2 + field)

  plot(y_pred)  

p <- graph$plot(data = "y", plotly=TRUE)
plot(y_pred, plotly = TRUE, p = p)

Using inlabru to fit models with replicates

We can also use our inlabru implementation to fit models with replicates. We will consider the same data that was generated above, where the number of replicates is 30.

For this implementation we will use the rspde_model_rep object.

We can now create the component, passing the vector with the indices of the replicates as the replicate argument. To obtain the auxiliary data, we will pass repl argument we use the function graph_data_rspde(), where we set it to .all, since we want all replicates. Further, we also pass the loc_name argument.

data_rspde_rep <- graph_data_rspde(rspde_model_rep, repl = ".all", loc_name = "loc")

We can now define the bru component formula, passing the repl as the replicate argument:

  cmp_rep <-
    y ~ -1 + field(loc, model = rspde_model_rep,
                              replicate = repl)

Now, we are ready to fit the model:

  rspde_bru_fit_rep <-
    bru(cmp_rep,
        data=data_rspde_rep[["data"]],
      options=list(
      family = "gaussian")
    )

We can obtain the estimates in the original scale with the rspde.result() function:

  result_bru_fit_rep <- rspde.result(rspde_bru_fit_rep, "field", rspde_model_rep)
  summary(result_bru_fit_rep)

##             mean         sd 0.025quant 0.5quant 0.975quant     mode
## std.dev 1.337860 0.03153180   1.269220 1.341100   1.390940 1.353590
## range   0.174413 0.00716215   0.159044 0.175042   0.186715 0.177525
## nu      0.729764 0.02390240   0.689765 0.727193   0.782398 0.718915

Let us compare with the true values of the parameters:

  result_bru_rep_df <- data.frame(
    parameter = c("std.dev", "range", "nu"),
    true = c(sigma, range, nu),
    mean = c(
      result_bru_fit_rep$summary.std.dev$mean,
      result_bru_fit_rep$summary.range$mean,
      result_bru_fit_rep$summary.nu$mean
    ),
    mode = c(
      result_bru_fit_rep$summary.std.dev$mode,
      result_bru_fit_rep$summary.range$mode,
      result_bru_fit_rep$summary.nu$mode
    )
  )
  print(result_bru_rep_df)

##   parameter true      mean      mode
## 1   std.dev 1.30 1.3378609 1.3535910
## 2     range 0.15 0.1744128 0.1775248
## 3        nu 0.80 0.7297645 0.7189153

We can also plot the posterior marginal densities with the help of the gg_df() function:

  posterior_df_bru_fit_rep <- gg_df(result_bru_fit_rep)

  ggplot(posterior_df_bru_fit_rep) + geom_line(aes(x = x, y = y)) + 
  facet_wrap(~parameter, scales = "free") + labs(y = "Density")

Let us now do prediction for observations of replicate 10. We start by building the data list with the prediction locations:

  data_prd_list_repl <- graph$get_mesh_locations(bru = TRUE,
                                            loc = "loc")
  data_prd_list_repl[["repl"]] <- rep(10, nrow(data_prd_list$loc))

Let us now obtain predictions for this replicate:

  y_pred <- predict(rspde_model_rep, cmp_rep, rspde_bru_fit_rep, 
                      newdata=data_prd_list_repl, ~field_eval(loc, replicate = repl))

We can now plot the predictions along with the observed values for replicate 10:

p <- plot(y_pred, plotly=TRUE)
graph_rep$plot(data = "y", group = 10, plotly=TRUE, p = p)

An example with a non-stationary model

Our goal now is to show how one can fit model with non-stationary \(\sigma\) (std. deviation) and non-stationary \(\rho\) (a range parameter). One can also use the parameterization in terms of non-stationary SPDE parameters \(\kappa\) and \(\tau\).

We follow the same structure as INLA. However, INLA only allows one to specify B.tau and B.kappa matrices, and, in INLA, if one wants to parameterize in terms of range and standard deviation one needs to do it manually. Here we provide the option to directly provide the matrices B.sigma and B.range.

The usage of the matrices B.tau and B.kappa are identical to the corresponding ones in inla.spde2.matern() function. The matrices B.sigma and B.range work in the same way, but they parameterize the stardard deviation and range, respectively.

The columns of the B matrices correspond to the same parameter. The first column does not have any parameter to be estimated, it is a constant column.

So, for instance, if one wants to share a parameter with both sigma and range (or with both tau and kappa), one simply let the corresponding column to be nonzero on both B.sigma and B.range (or on B.tau and B.kappa).

 pems_graph <- metric_graph$new(edges = pems$edges, longlat=TRUE)
 df_data_pems = data.frame(y = pems$Y, edge_number = pems$PtE[,1], distance_on_edge = pems$PtE[,2])
 pems_graph$add_observations(data = df_data_pems, normalized=TRUE)
 pems_graph$prune_vertices()
 pems_graph$build_mesh(h=0.1)

summary(pems_graph)

## A metric graph object with:
## 
## Vertices:
##   Total: 347 
##   Degree 1: 11;  Degree 2: 16;  Degree 3: 315;  Degree 4: 5; 
##   With incompatible directions:  17 
## 
## Edges: 
##   Total: 504 
##   Lengths: 
##       Min: 0.01040218  ; Max: 7.677232  ; Total: 470.7559 
##   Weights: 
##       Min: 1  ; Max: 1 
##   That are circles:  0 
## 
## Graph units: 
##   Vertices unit:  degrees  ; Lengths unit:  km 
## 
## Longitude and Latitude coordinates:  TRUE
##   Which spatial package:  sp 
##   CRS:  +proj=longlat +datum=WGS84 +no_defs
## 
## Some characteristics of the graph:
##   Connected: TRUE
##   Has loops: FALSE
##   Has multiple edges: TRUE
##   Is a tree: FALSE
##   Distance consistent: FALSE
##   Has Euclidean edges: FALSE
## 
## Computed quantities inside the graph: 
##   Laplacian:  FALSE  ; Geodesic distances:  TRUE 
##   Resistance distances:  FALSE  ; Finite element matrices:  FALSE 
## 
## Mesh: 
##   Max h_e:  0.09998277  ; Min n_e:  0 
## 
## Data: 
##   Columns:  y 
##   Groups:  None 
## 
## Tolerances: 
##   vertex-vertex:  1e-07 
##   vertex-edge:  1e-07 
##   edge-edge:  0

cov_pos <- sin(pems_graph$mesh$VtE[,2] * pi)

 B.sigma = cbind(0, 1, 0, cov_pos)
 B.range = cbind(0, 0, 1, cov_pos)

fit <- graph_lme(y ~ 1, graph = pems_graph, model = list(type = "WhittleMatern", 
                    B.sigma = B.sigma, B.range = B.range, fem = TRUE), improve_hessian = TRUE)

## Warning in rSPDE::rspde_lme(formula = formula, loc = cbind(df_data[[".edge_number"]], : optim method L-BFGS-B failed to provide a positive-definite Hessian. Another optimization method was used.

summary(fit)

## 
## Latent model - Generalized Whittle-Matern
## 
## Call:
## graph_lme(formula = y ~ 1, graph = pems_graph, model = list(type = "WhittleMatern", 
##     B.sigma = B.sigma, B.range = B.range, fem = TRUE), improve_hessian = TRUE)
## 
## Fixed effects:
##             Estimate Std.error z-value Pr(>|z|)    
## (Intercept)   49.865     3.054   16.33   <2e-16 ***
## 
## Random effects:
##         Estimate Std.error z-value
## alpha     2.0285    0.1122  18.077
## Theta 1   2.0788    1.3289   1.564
## Theta 2   1.3878    1.0654   1.303
## Theta 3   1.1100    3.6545   0.304
## 
## Measurement error:
##          Estimate Std.error z-value
## std. dev  1.95681   0.05681   34.45
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Log-Likelihood:  -1209.895 
## Number of function calls by 'optim' = 451
## Optimization method used in 'optim' = Nelder-Mead
## 
## Time used to:     fit the model =  11.95274 secs 
##   compute the Hessian = 6.12453 secs

est_range <- exp(B.range[,-1]%*%fit$coeff$random_effects[2:4])
pems_graph$plot_function(X = est_range, vertex_size = 0)

est_sigma <- exp(B.sigma[,-1]%*%fit$coeff$random_effects[2:4])
pems_graph$plot_function(X = est_sigma, vertex_size = 0)

rspde_object_ns <- rSPDE::spde.matern.operators(graph = pems_graph,
                                                parameterization = "matern",
                                                B.sigma = B.sigma,
                                                B.range = B.range,
                                                theta = fit$coeff$random_effects[2:4],
                                                nu = fit$coeff$random_effects[1] - 0.5)

est_cov_matrix <- rspde_object_ns$covariance_mesh()
est_std_dev <- sqrt(diag(est_cov_matrix))

pems_graph$plot_function(X = est_std_dev, vertex_size = 0)

rspde_model_nonstat <- rspde.metric_graph(pems_graph,
  B.sigma = B.sigma,
  B.range = B.range,
  parameterization = "matern") 

 data_rspde_bru_ns <- graph_data_rspde(rspde_model_nonstat, loc_name = "loc")

cmp_nonstat <-
  y ~ -1 + Intercept(1) + field(loc,
    model = rspde_model_nonstat
  )


rspde_fit_nonstat <-
  bru(cmp_nonstat,
    data = data_rspde_bru_ns[["data"]],
    family = "gaussian",
    options = list(verbose = FALSE)
  )

summary(rspde_fit_nonstat)

## inlabru version: 2.10.0
## INLA version: 23.05.30-1
## Components:
## Intercept: main = linear(1), group = exchangeable(1L), replicate = iid(1L)
## field: main = cgeneric(loc), group = exchangeable(1L), replicate = iid(1L)
## Likelihoods:
##   Family: 'gaussian'
##     Data class: 'metric_graph_data', 'list'
##     Predictor: y ~ .
## Time used:
##     Pre = 0.169, Running = 37, Post = 0.23, Total = 37.4 
## Fixed effects:
##             mean    sd 0.025quant 0.5quant 0.975quant   mode kld
## Intercept 50.721 3.029      44.57   50.757     56.661 50.754   0
## 
## Random effects:
##   Name     Model
##     field CGeneric
## 
## Model hyperparameters:
##                                          mean    sd 0.025quant 0.5quant 0.975quant  mode
## Precision for the Gaussian observations 0.020 0.003      0.016    0.020      0.025 0.020
## Theta1 for field                        2.843 0.226      2.388    2.844      3.287 2.852
## Theta2 for field                        1.890 0.222      1.440    1.891      2.323 1.903
## Theta3 for field                        0.531 0.354     -0.164    0.530      1.245 0.516
## Theta4 for field                        0.662 0.347     -0.024    0.661      1.354 0.657
## 
## Deviance Information Criterion (DIC) ...............: 2301.67
## Deviance Information Criterion (DIC, saturated) ....: 426.27
## Effective number of parameters .....................: 100.37
## 
## Watanabe-Akaike information criterion (WAIC) ...: 2296.02
## Effective number of parameters .................: 77.80
## 
## Marginal log-Likelihood:  -1235.98 
##  is computed 
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')

result_fit_nonstat <- rspde.result(rspde_fit_nonstat, "field", rspde_model_nonstat)
summary(result_fit_nonstat)

##                   mean       sd 0.025quant 0.5quant 0.975quant     mode
## Theta1.matern 2.842650 0.225720   2.387640 2.843580    3.28719 2.851830
## Theta2.matern 1.889610 0.222284   1.440090 1.891300    2.32255 1.903020
## Theta3.matern 0.531426 0.353636  -0.164039 0.529808    1.24505 0.515564
## nu            1.311280 0.152284   0.993453 1.318900    1.58598 1.335790

posterior_df_fit <- gg_df(result_fit_nonstat)

ggplot(posterior_df_fit) + geom_line(aes(x = x, y = y)) + 
facet_wrap(~parameter, scales = "free") + labs(y = "Density")